The idea of -almost convergence (briefly, -convergence) has been recently introduced and studied by Mohiuddine and Alotaibi (2014). In this paper first we define a norm on such that it is a Banach space and then we define and characterize those four-dimensional matrices which transform -convergence of double sequences into -convergence. We also define a -core of and determine a Tauberian condition for core inclusions and core equivalence.

1. Background, Notations, and Preliminaries

We begin by recalling the definition of convergence for double sequences which was introduced by Pringsheim [1]. A double sequence is said to be to in the Pringsheim's sense (or -convergent to ) if for given there exists an integer such that whenever . We will write this as where and are tending to infinity independent of each other. We denote by the space of -convergent sequences.

We say that a double sequence is if Denote by the space of all bounded double sequences.

If a double sequence in and is also -convergent to , then we say that it is boundedly -convergent to (or, BP-convergent to ). We denote by the space of all boundedly -convergent double sequences. Note that .

We remark that, in contrast to the case for single sequences, a -convergent double sequence need not be bounded.

Let be a four-dimensional infinite matrix of real numbers for all and a space of double sequences. Let be a double sequences space, converging with respect to a convergence rule . Define Then, we say that a four-dimensional matrix maps the space into the space if and is denoted by .

The idea of almost convergence of Lorentz [2] is narrowly connected with the limits of Banach (see [3]) as follows. A sequence in is almost convergent to if all of its Banach limits are equal, where denotes the space of all bounded sequences. Mohiuddine [4] applies this concept to established some approximation theorems for sequence of positive linear operator. Móricz and Rhoades [5] extended the notion of almost convergence from single to double sequences as follows.

A double sequence of real numbers is said to be almost convergent to a number if For more details on almost convergence for single and double sequences, one can refer to [613].

The two-dimensional analogue of Banach limit has been defined by Mursaleen and Mohiuddine [14] as follows. A linear functional on is said to be Banach limit if it has the following properties: if (i.e., for all ),, where with for all ,,where the shift operators , , and are defined by Denote by the set of all Banach limits on . Note that if (B) holds, then we may also write . A double sequence is said to be almost convergent to a number if for all .

Let and be two nondecreasing sequences of positive reals and each tending to such that , , , , and is called the double generalized de la Valée-Poussin mean, where and . We denote the set of all and type sequences by using the symbol .

Quite recently, Mohiuddine and Alotaibi [15] presented a generalization of the notion of almost convergent double sequence with the help of de la Vallée-Poussin mean and called it -almost convergent. In the same paper, they also defined and characterized some four-dimensional matrices. For more details on double sequences, four-dimensional matrices, and other related concepts, one can refer to [1626].

A double sequence of reals is said to be -almost convergent (briefly, -convergent) [15] to some number if , where Denote by the space of all almost convergent sequences . Note that .

We remark that if we take and , then the notion of -almost convergence coincides with the notion of almost convergence for double sequences due to Móricz and Rhoades [5].

2. -Matrices

We will assume throughout this paper that the limit of a double sequence means limit in the Pringsheim sense. We define the following matrix classes and establish interesting results.

Definition 1. A four-dimensional matrix is said to be -almost regular if for all with , and one denotes this by .

Definition 2. A matrix is said to be of class if it maps every -convergent double sequence into -convergent double sequence; that is, for all . In addition, if , then is and, in symbol, one will write .

Now we define the norm on as follows.

Theorem 3. is a Banach space normed by

Proof. It can be easily verified that (8) defines a norm on . We show that is complete. Now, let be a Cauchy sequence in . Then for each , is a Cauchy sequence in . Therefore (say). Put ; given there exists an integer , say, such that, for each , Hence then, for each , , , and , , we have Taking limit , we have for and for each of Now for fixed , the above inequality holds. Since for fixed , , we get uniformly in , . For given , there exist positive integers , such that for , and for all , . Here , are independent of , but depend upon . Now by using (12) and (14), we get for , and for all , . Hence and is complete.

Now we characterize the matrix class as well as . Let be the subspace of such that , uniformly in ; that is Note that every can be written as where uniformly in , and with for all .

Theorem 4. A matrix if and only if (A),(A),(A),where is the shift operator.

Necessity. Let . We know that , so we have . Hence the necessity of (A) follows. Since , then . This is equivalent to that is, (A) holds. For each , we have because for all Banach limit . Hence ; that is, (A) holds.
Sufficiency. Let conditions (A)–(A) hold and . Then where , , uniformly in and with for all . Taking -transform in (20), we obtain If , then by (A) we have . Since by (A), is bounded linear operator on , we get . This yields . Now from condition (A) and (21), . Therefore .

Corollary 5. A matrix if and only if conditions (A) and (A) with and (A) hold.

3. Some Core Theorems

The core or Knopp core of a real number single sequence is the closed interval (see [27, 28]). In 1999, Patterson [29] extended the Knopp core from single sequences to double sequences and called it Pringsheim core (shortly, -core) which is given by . In the recent past, the -core and -core for double sequences have been defined and studied by Mursaleen and Edely [30] and Mursaleen and Mohiuddine [31, 32], respectively, while the -core for single sequences is given by Mishra et al. [33]. In 2011, Kayaduman and Çakan [34] presented the concept of Cesáro core of double sequences.

We define the following sublinear functional on : Then we define the -core of a real-valued bounded double sequence to be the closed interval .

Since every -convergent double sequence is -convergent, we have where , and hence it follows that for all .

Theorem 6. For every , if and only if(C) is -regular,(C),where

Necessity. Suppose that (24) holds for all . One obtains that is, If , then that is, Therefore is -regular. This yields the necessity of (C).
Now, with the help of Lemma 2.1 of [35], there is a double sequence such that and If a double sequence defined by then where This yields the necessity of (C).
Sufficiency. We know that . Following the lines of Theorem 2 of [31] for translation mapping, one obtains For any , we have Taking infimum over , we obtain Thus Since , we can write where , since is -regular. Operating to (38), one obtains By -almost regularity, we have From the definition of , we get uniformly in . Also Therefore we obtain from (40) that uniformly in . Equations (37) and (43) give that that is, As , one obtains .

Note that . This motivates us to prove the following result by adding a condition to get a more general result.

Theorem 7. For , if holds, then .

Proof. By the definition of -core and -core, we have to show that . Let . Then, for given , for all and for large it follows from the definition of that We can write Since (46) holds, for given , we get that for all . Thus we have Equation (49) yields Taking in (48) and using (51), one obtains . Since is arbitrary, we obtain .

Corollary 8. If (46) holds and is -convergent, then is convergent.

Finally, we define the concepts of -almost uniformly positive and -almost absolutely equivalent and establish a theorem related to these concepts.

Definition 9. A matrix is said to be -almost uniformly positive, denoted by -uniformly positive, if

Definition 10. Let and be two -regular matrices and Then and are said to be -almost absolutely equivalent, denoted by -absolutely equivalent, on whenever ; that is, either and both tend to the same -limit or neither of them tends to a -limit, but their difference tends to -limit zero.

Before proceeding further, first we state the following lemma which we will use to our next result.

Lemma 11. For , if , then .

Proof of the lemma is straightforward and thus omitted.

Theorem 12. Let be a -regular matrix. Then, for all if and only if there is a -regular matrix such that is -uniformly positive and -absolutely equivalent with on .

Proof. Let there be a -regular matrix such that is -uniformly positive and -absolutely equivalent with on . Then, by (53) and -absolutely equivalent of and , we have uniformly in . Now, by Lemma 11, for all . By Theorem 6, we have , since is arbitrary.
Conversely, let for all . Then by Theorem 6, is -uniformly positive. Now we define a matrix as for all , , , . Then it is easy to see that is -regular since is -regular, and Further Since is -regular, we have by (57) that Thus is -uniformly positive. Further, it follows from (56) that and are -absolutely equivalent.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (130-265-D1435). The authors, therefore, acknowledge with thanks DSR technical and financial support.